## Advances in Differential Equations

- Adv. Differential Equations
- Volume 6, Number 4 (2001), 385-418.

### Semiclassical limit for a quasilinear elliptic field equation: one-peak and multipeak solutions

Marino Badiale, Vieri Benci, and Teresa D'Aprile

#### Abstract

This paper deals with the existence of one-bump and multibump solutions for the following nonlinear field equation: $$-\Delta u+V(h x)u-\Delta_{p}u+ W'(u)=0$$ where $u:\mathbb R^{N}\rightarrow\mathbb R^{N+1},$ $N\geq 2,$ $p>N,$ $h>0,$ the potential $V$ is positive and $W$ is an appropriate singular function. Existence results are established provided that $h$ is sufficiently small, and we find solutions exhibiting a concentration behaviour in the semiclassical limit (i.e., as $h\rightarrow 0^{+}$) at any prescribed finite set of local minima, possibly degenerate, of the potential. Such solutions are obtained as local minima for the associated energy functional. No restriction on the global behaviour of $V$ is required except that it is bounded below away from zero. In the proofs of these results we use a variational approach, and the method relies on the study of the behaviour of sequences with bounded energy, in the spirit of the concentration-compactness principle.

#### Article information

**Source**

Adv. Differential Equations, Volume 6, Number 4 (2001), 385-418.

**Dates**

First available in Project Euclid: 2 January 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1357140605

**Mathematical Reviews number (MathSciNet)**

MR1798491

**Zentralblatt MATH identifier**

1012.35026

**Subjects**

Primary: 35J50: Variational methods for elliptic systems

Secondary: 35A05 35B05: Oscillation, zeros of solutions, mean value theorems, etc.

#### Citation

Badiale, Marino; Benci, Vieri; D'Aprile, Teresa. Semiclassical limit for a quasilinear elliptic field equation: one-peak and multipeak solutions. Adv. Differential Equations 6 (2001), no. 4, 385--418. https://projecteuclid.org/euclid.ade/1357140605